Credibility Theory and Kalman Filtering with Extensions

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CREDIBILITY THEORY AND K A L W N FILTERING WITH EXTENSIONS

R. K. Mehra

November 19 75

Research Memoranda are informal publications relating to ongoing or projected areas o f research at IIASA. The views expressed are those of the author, and do not necessarily reflect those of IIASA.

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Credibility Theory and Kalman Filtering with Extensions

R. K. Mehra

*

Introduction

Credibility theory developed in actuarial mathematics involves the estimating of risk premiums based on the collective and

individual risk data. Though relatively old in origin [ I 51

,

it has only recently been put on a sound mathematical foundation

[1,2,10]. A number of extensions of the theory have also appeared recently [3,7] under the titles of "multidimensional credibility theory," "time-inhomogeneous credibility theory" and "evolutionary credibility theory."

In this paper it is shown that the estimate and prediction problems considered in credibility theory are similar to those considered in Kalman filtering theory. The state vector of the risk model consists of the average risk variables such as the number of claims, cost of claims, etc. for a particular risk 0 from the collective of risks 0 . The observed data consists of the risk variables for the collective and for the individual risks. It is required to predict the values of risk variables in the next time period based on this data and to adjust the individual premiums based on claims experience in such a way as to converge to their true values.

The application of Kalman filtering, innovation theory and maximum likelihood estimation [12] gives results more general than those obtained before. It also opens up the possibility for considering more general models and testing their validity on real data. On the other hand, credibility theory provides

certain specialized results which have not been fully appreciated in the Kalman filtering literature. In particular, it has been shown in credibility theory that the same prediction formulae hold under a variety of measurement noise distribution functions such as exponential, poisson, binomial, etc. [ G I . ^{Thus, the} connection between credibility theory and Kalman filtering is significant and may be expected to lead to further developments in both areas.

*

Harvard university, Cambridge, Plassachusetts, and IIASA, Laxenburg, Austria.

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I. Problem Formulation

Let 8 denote the risk parameter that varies from one individual to the other. Let U(8) be the prior distribution function of 6 over the whole population of individuals.

Let ~11~2,...,<t,...tSn be the experience observation vectors on an individual with a specific 8. Prediction of 'n+l based on the population statistics and the experience observations is required.

Assume that St(8) is normally distributed with mean mt (8) and covariance Ct (8) for a specific individual. We may write

where v (8) is a white noise sequence, normally distributed t

with zero mean and covariance ~ ( 8 ) for a given 8. Thus

<1(8)I...,<n(0) are conditionally independent. Further let the expected values of mt(8) and Ct(8) over the population of all 8 be mt and Et:

and

The Kalman filter formulation [I51 requires setting up a state equation and a measurement equation. Equation (1) is essentially the measurement equation with mt(8) representing the "signal" part and vt(8) representing the "noise" part.

The evolution of the "signal" part is described by the state equation, which in the most general form can be taken to be,

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Here x denotes the state of the system which can have t

dimension greater or less than mt(8) and ut is a normally distributed white noise sequence with known mean (say zero) and covariance Qt.

The model (2)-(3) is perhaps too general for the "credibility problem," but it is stated here since the Kalman filter is valid for this general model. The model (2)

-

⁽³⁾^{may be}

specialized to various cases considered in Jewel1 [ 7 1 as follows:

i) Time-Homogeneous Risks Submodel: Let F t = I

,

^{u = O}_t-

,

^Ht⁼ ^I

then

Also since m (8) is normally distributed with mean m and co- o

variance D,

Furthermore Et is asswed constant ( = El.

ii) Time-Dependent Risk Model: ut : 0, H = I, so that t

n (8) = Ftnt (6 )

.

The other quantities sane as above t+l

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iii) Evolutionary Risk Model: Ft = I, Ht = I, Gt = I so that

with xo, Po, and Et the same as in i). Now we state the Kalman filter equations and show how they simplify to the

"credibility theory" results.

11. Kalman Filtering: [5,8,9]

Let x

t/.r denote the best estimate of xt based on the observation

5,)

and let Pt,, be its covariance. From least squares theory,

A

Then the "filtered" estimates x

t

I

^tand the "predicted" estimates

A

X t+l

1

^tcan be obtained recursively as follows:

Equations (11)-(12) may also be written as

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which w i l l be u s e f u l i n r e l a t i n g t o c r e d i b i l i t y t h e o r y . E q u a t i o n s ( 1 1 ) - ( 1 3 ) may b e combined i n t o a s i n g l e r e c u r s i v e e q u a t i o n f o r P

tit

^{O r}P t l t - l t e - g -

E q u a t i o n ( 1 6 ) i s a d i s c r e t e - t i m e R i c c a t i e q u a t i o n w h i c h h a s b e e n s t u d i e d e x t e n s i v e l y i n t h e c o n t r o l l i t e r a t u r e [ 5 1 . The

s e q u e n c e S t

-

Htxt t-l known a s t h e " i n n o v a t i o n s e q u e n c e " i s a G a u s s i a n w h i t e n o i s e s e q u e n c e [8]. W e now c o n s i d e r t h e f i r s t s p e c i a l case o f c r e d i b i l i t y t h e o r y .

i ) Time-Homogeneous R i s k Model : S u b s t - i t u t i n g f o r F t ' Ht e t c . , i n E q u a t i o n ( 9 ) - ( 1 6 ) ,

Thus

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By induction,

where

1 -1 1

Zn = (%ED

+

^I)-'

^{= E ~ n}

(Credibility Factor). (20) Equation (19) is the same as Equation (4.26) of Jewel1 [ 7 1 .

Kalman filter results for models ii) and iii) also lead

to parallel results in credibility theory. We do not pursue this approach here since from a computational view point, the recursive Equations (9)-(13) are easier to solve.

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111. Extensions

In credibility theory as well as in several applications of Kalman filtering, the matrices F, G, H, Q and R contain unknown parameters. This problem has been studied extensively in the control literature [11,12,13,14] and computer programs exist for obtaining least squares and maximum likelihood

estimates of these parameters. The need for such methods in Credibility theory has already been indicated by ~Ghlmann [ 4 1 .

An objection to the use of Kalman filtering for credibility theory is the assumption of normality. In the non-normal add- itive noise case, the Kalman filter still provides optimal linear least squares estimates. Several nonlinear estimating techniques also exist (Jazwinski [5] ) and may be used for

problems in credibility theory. Another approach is to trans- form the data in such a way that the transformed variables have normal distributions. For example, log- normal distributions can

be handled in this fashion.

Conclusions

A connection has been established between problems and results from two different fields viz: Kalman filtering

and credibility theory of actuarial mathematics. A number of extensions of credibility theory are indicated based on results from Kalman filtering, innovation theory, and maximum likelihood identification.

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R e f e r e n c e s

B a i l e y , A . L . "A G e n e r a l i z e d Theory o f C r e d i b i l i t y . "

P r o c e e d i n g s o f t h e C a s u a l t y A c t u a r i a l S o c i e t y , Vol. 32.

1945, pp. 13-20.

B a i l e y , A.L. " C r e d i b i l i t y P r o c e d u r e s , L a p l a c e ' s G e n e r a l i - z a t i o n o f B a y e s ' R u l e and t h e C o m b i n a t i o n o f

C o l l a t e r a l Knowledge w i t h Observed D a t a . " P r o c e e d i n g s o f t h e C a s u a l t y A c t u a r i a l S o c i e t y , Vol. 3 7 . 1950, pp. 7-23.

Buhlmann, H . " C r e d i b i l i t y P r o c e d u r e s . " S i x t h B e r k e l e y

Symposium. 1971, pp. 515-525.

~ u h l m a n n , H . M a t h e m a t i c a l Methods i n R i s k T h e o r y , N e w York, S p r i n g e r - V e r l a g , 1970.

J a z w i n s k i , A.H. S t o c h a s t i c P r o c e s s e s a n d F i l t e r i n g T h e o r y , N e w York, Academic Press, 1970.

J e w e l l , W.S. " C r e d i b l e Means a r e E x a c t B a y e s i a n f o r S i m p l e E x p o n e n t i a l F a m i l i e s . " ORC-73-21. O p e r a t i o n s R e s e a r c h C e n t e r . U n i v e r s i t y o f C a l i f o r n i a , B e r k e l e y , O c t o b e r 1973 ( t o a p p e a r i n ASTIN B u l l e t i n )

.

J e w e l l , W.S. "Model V a r i a t i o n s i n C r e d i b i l i t y T h e o r y . "

ORC-74-25. O p e r a t i o n s R e s e a r c h C e n t e r . U n i v e r s i t y o f C a l i f o r n i a , B e r k e l e y , August 1974.

K a i l a t h , T. "An I n n o v a t i o n s Approach t o L e a s t - S q u a r e s

E s t i m a t i o n , P a r t 1 . " IEEE T-AC, AC-13 (December 1 9 6 8 ) . Kalman, R.E. "New Methods and R e s u l t s i n L i n e a r P r e d i c t i o n

and F i l t e r i n g T h e o r y . " P r o c . Symp. o n Engg. Appl. o f Random F u n c t i o n T h e o r y a n d P r o b a b i l i t y . N e w York, W i l e y , 1961.

Mayerson, A.L. "A ~ a y e s i a n V i e w o f C r e d i b i l i t y . "

P r o c e e d i n g s o f t h e C a s u a l t y A c t u a r i a l S o c i e t y , V o l . 51.

1964, pp. 85-104.

Mehra, R . K . " A p p r o a c h e s t o A d a p t i v e F i l t e r i n g . " IEEE T-AC ( O c t o b e r 1 9 7 2 ) .

Mehra, R . K . " ~ d e n t i f i c a t i o n i n C o n t r o l and E c o n o m e t r i c S y s t e m s , S i m i l a r i t i e s and D i f f e r e n c e s . " A n n a l s o f Economic and S o c i a l Measurements ( J a n u a r y 1 9 7 4 )

.

Mehra, R . K . " I d e n t i f i c a t i o n o f S t o c h a s t i c L i n e a r Dynamic S y s t e m s U s i n g Kalman F i l t e r R e p r e s e n t a t i o n . " A I A A J o u r n a l ( J a n u a r y 1971 )

.

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[I 4 1 Mehra, R . K . "On L i n e I d e n t i f i c a t i o n o f L i n e a r Dynamic Systems w i t h A p p l i c a t i o n s t o Kalman F i l t e r i n g . "

I E E E T-AC, Vol. AC-16 No. 1 ( F e b r u a r y 1971 )

.

[I51 Whitney, A.W. "The Theory of E x p e r i e n c e R a t i n g . " Proceed- i n g s of t h e C a s u a l t y A c t u a r i a l S o c i e t y , Vol. 4. 1918, pp. 274-292.